已知数列\(\{a_{n}\}\)是递增等比数列，且\(a_{3}=4\)，\(a_{2}+a_{4}=10\)，\(S_{n}\)为等差数列\(\{b_{n}\}\)的前\(n\)项和，且\(b_{1}=a_{1}\)，\(S_{2}=a_{2}+1.\)
\((1)\)求数列\(\{a_{n}\}\)，\(\{b_{n}\}\)的通项公式；
\((2)\)若\(c_{n}=a_{n}b_{n}\)，求数列\(\{c_{n}\}\)的前\(n\)项和\(T_{n}.\)